Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning

Abstract

We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series. The FLM uses a simple feedforward structure with cosine activation functions to learn the frequencies, amplitudes, and phase shifts of the series as trainable parameters. This design allows the model to create a problem-specific spectral basis adaptable to both periodic and nonperiodic functions. Unlike previous Fourier-inspired NN models, the FLM is the first architecture able to represent a multidimensional Fourier series with a complete set of basis functions in separable form, doing so by using a standard Multilayer Perceptron-like architecture. A one-to-one correspondence between the Fourier coefficients and amplitudes and phase-shifts is demonstrated, allowing for the translation between a full, separable basis form and the cosine phase-shifted one. Additionally, we evaluate the performance of FLMs on several scientific computing problems, including benchmark Partial Differential Equations (PDEs) and a family of Optimal Control Problems (OCPs). Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.

Reproductions